September 4, 2007
Number Theory Problems (for tonight)
1. Let 1, 4, … and 9, 16, … be two arithmetic progressions. The set S is the union of the first 2007 terms of each sequence. How many distinct numbers are in S ?
2. What is the largest positive integer n for which n3 + 100 is divisible by n + 10 ?
3. A positive integer is written on each face of a cube. To each vertex, we assign the product of the numbers written on the three faces meeting at that vertex. The sum of the numbers assigned to the vertices is 1001. What is the sum of the numbers written on the faces ?
4. A positive integer k greater than 1 is given. Prove that there exist a prime number p and a strictly increasing sequence of integers a1, a2, a3, … such that
p + ka1, p + ka2, p + ka3, …
are all primes.
5. Given a positive integer n, let p(n) be the product of the nonzero digits of n (or n itself if n has only one digit). For example:
p(3) = 3 p(30) = 3 P(641) = 24
S = p(1) + p(2) + p(3) + … + p(999).
What is the largest prime factor of S ?
6. The number 27000001 is the product of four distinct primes. What are they?
7. (a) Can the sum of six consecutive squares be a square?
(b) Can the sum of eleven consecutive squares be a square?
8. For any positive integer n, the expression s(n) means the sum of all the positive divisors of n. For example, s(6) = 1 + 2 + 3 + 6 = 12. Show that for any n,
s(1) + s(2) + s(3) + … + s(n) ≤ n2.